No-feedback card guessing for dovetail shuffles

Abstract

We consider the following problem. A deck of $2n$ cards labeled consecutively from 1 on top to $2n$ on bottom is face down on the table. The deck is given k dovetail shuffles and placed back on the table, face down. A guesser tries to guess at the cards one at a time, starting from top. The identity of the card guessed at is not revealed, nor is the guesser told whether a particular guess was correct or not. The goal is to maximize the number of correct guesses. We show that, for $k \geq 2 \log_2 (2n) + 1$, the best strategy is to guess card 1 for the first half of the deck and card $2n$ for the second half. This result can be interpreted as indicating that it suffices to perform the order of $\log_2(2n)$ shuffles to obtain a well-mixed deck, a fact proved by Bayer and Diaconis. We also show that if $k = c \log_2 (2n)$ with $1 < c < 2$, then the above guessing strategy is not the best.